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\title{\huge Cassinian Oval \footnote{\large This file is from the 3D-XploreMath project. 
\hfil\break Please see http://www.math.uci.edu/$\sim$palais/  or http://3d-xplormath.org/}}
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\LARGE


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The Cassinian Ovals (or Ovals of Cassini) were first studied in 1680 by Giovanni 
Domenico Cassini (1625--1712, aka Jean-Dominique Cassini) as a model for the orbit of the Sun 
around the Earth.

A Cassinian Oval is any plane curve that is the locus of all points $P$ such that the
product of the distances of $P$ from two fixed points has some fixed value, 
 that is: $\overline{P\,F_1} \, \overline{P\,F_2} = c$ , where
$F_1$ and $F_2$ are two points and $c$ is a constant. Note the analogy with
the definition of an ellipse (where product is replaced by sum). As with the 
ellipse, the two points $F_1$ and $F_2$ are called the {\it foci\/} of the oval.
If we locate the origin of our coordinates at the midpoint of the two foci and choose
for the $x$-axis the line joining them, then the foci will have the coordinates 
$(a,0)$ and $(-a,0)$. Following convention we will let $b$ denote the square 
root of $c$, and then the condition for a point $P = (x,y)$ to lie on the oval 
becomes:
\[
\sqrt{(x-a)^2+y^2} \sqrt{(x+a)^2+y^2}=b^2
\]
and on squaring the two sides, we end up with the following quartic polynomial 
equation for the Cassinian Oval:
\[
((x-a)^2+y^2) ( (x+a)^2+y^2)=b^4
\]

When $b$ is less that half the distance $2 a$ between the foci, i.e., $b/a < 1$,
 there are two branches of the curve.  When $a=b$, the curve has the shape 
of a figure eight and is known as the Lemniscate of Bernoulli.

The following image shows a family of Cassinian Ovals with $a = 1$ and 
several different values of $b$.
\bigskip

\centerline{\includegraphics{cassinianOval.png}}

In 3D-XplorMath, you can change the value of parameter b in the menu Settings $\rightarrow$ SetParameters. An animation of varying values of b can be seen from the menu Animate$\rightarrow$ Color Morph.

Bipolar equation: $r_1 r_2 = b^2$

Polar equation: $r^4 + a^4 - 2 r^2 a^2 \cos(2 \theta) = b^4$

The parametric formula for Cassinian oval is $\sqrt{M/2} (\cos(t), \sin(t))$, where M is
 \[2 a^2 \cos(2 t) + 2 \sqrt{(-a^4+b^4) + a^4 (\cos(2 t))^2},
 \]
with $0 < t \leq 2 \pi$, and $a < b$. This parametrization only generates parts of the curve when $a > b$.

 By default 3D-XplorMath shows how the product definition of the Cassinian ovals leads to a ruler and
 circle construction based on the following circle theorem about products of segments:
 
\centerline{\includegraphics[width=5  in]{CircleProduct.png}}    %\vglue -15pt

{\bf  Cassinian Ovals as sections of a Torus}

Let c be the radius of the generating circle and $d$ the distance from the center of the tube to the directrix of the torus. The intersection of a plane $c$ distant from the torus' directrix is a Cassinian oval, with $a = d$ and 
$b^2 = \sqrt{4} c d$, where $a$ is half of the distance between foci, and $b^2$ is the constant  product of distances.

Cassinian ovals with a large value of $b^2$  approch a circle, and the corresponding torus is one such that the tube radius is larger than the center to directrix, that is, a self-intersecting torus without the hole. This surface also approaches a sphere.

Note that the two tori in the figure below are not identical. 

Arbitrary vertical slices of a torus are called Spiric Sections. In general they are {\it not\/} Cassinian ovals.


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\centerline{\includegraphics{cassinianOvalCut1.png}}	
\centerline{\includegraphics{cassinianOvalCut2.png}}	

Proof outline: Start with the equation of a torus $(\sqrt{x^2 + y^2} - d)^2 + z^2 = c^2$. Eliminate the square root and regroup to one side. Substitute $d=a$ and $c = b^2/(\sqrt{4}*a)$. Now do the same with Cassinian oval implicit equation $\sqrt{(x-a)^2+y^2} \sqrt{(x+a)^2+y^2}=b^2$. One sees that, up to a scale-factor and a rotation, the two equations match.

Curves that are the locus of points the product of whose distances from n points is constant are discussed on pages 60--63 of Visual Complex Analysis by Tristan Needham.

HK \& XL.

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